Problem:

(i) If A + B +C = n \pi , show that \sin 2A + \sin 2B + \sin 2C = (-1)^{n-1} 4 \sin A \sin B \sin C
(ii) Let triangles ABC and DEF be inscribed in the same circle. If the triangles are of equal perimeter, then prove that \sin A + \sin B + \sin C = \sin D + \sin E + \sin F
(iii) State and prove the converse of (ii) above

Discussion:

(i) We know transformation formula from trigonometry \sin x + \sin y = 2 \sin \dfrac{x+y}{2} \cos \dfrac {x-y}{2}

Hence \sin 2A + \sin 2B + \sin 2C = 2 \sin \dfrac{2A+2B}{2} \cos \dfrac {2A-2B}{2} + sin 2C = 2 \sin (A+B) \cos (A-B) + \sin 2C

Now we know that A + B = n \pi - C \Rightarrow \sin (A+B) = \sin (n \pi - C) = (-1)^{n-1} \sin C

So 2\sin (A+B) \cos (A-B) + \sin 2C = 2(-1)^{n-1} \sin C \cos (A-B) + 2 \sin C \cos C = 2(-1)^{n-1} \sin C \cos (A-B) + 2 \sin C \cos (n\pi -(A+B))

= 2(-1)^{n-1} \sin C \cos (A-B) + 2 (-1)^n \sin C \cos (A+B)

= 2(-1)^{n-1} \sin C (\cos (A-B) - \cos (A+B))

= 4(-1)^{n-1} \sin C \sin A \sin B

(ii)

Since the two triangles are inscribed in the same circle, they must have the same circumradius. Let the common circumradius be R. If a, b, c, d, e, f be the sides opposite to the sides BC, CA, AB, EF, DF, DE respectively, then using the rule of sines we can say,

\dfrac{\sin A}{a} = \dfrac { \sin B }{ b} = \dfrac {\sin C }{c} = \dfrac {1} {2R} and

\dfrac{\sin D}{d} = \dfrac { \sin E }{ e} = \dfrac {\sin F }{f} = \dfrac {1} {2R}

Hence \sin A = \dfrac {a}{2R}, sin B = \dfrac{b}{2R}, \sin C = \dfrac {c}{2R} \Rightarrow \sin A + \sin B + \sin C = \dfrac {a+b+c}{2R}

Similarly \sin D + \sin E + \sin F = \dfrac {d + e + f}{2R}

As the perimeter of the triangle are equal, hence a+b+c = d+e+f. This implies \sin A + \sin B + \sin C = \sin D + \sin E + \sin F

(iii)

We apply the sine rule in reverse order to get the converse.

Chatuspathi:

  • What is this topic: Property of triangles
  • What are some of the associated concept: Rule of sines
  • Where can learn these topics: Cheenta I.S.I. & C.M.I. course, discusses these topics in the ‘Trigonometry Module’ module.
  • Book Suggestions: Trigonometry by S.L. Loney